How do I apply the mirror modifier properly on animated geometry?
What do you use for animation? Armature, shape keys, or setting keyframes. Because it really depends on which one you use.


Armature (with bones)


Check out the ORDER of modifiers. If your Armature modifier is below your mirror modifier, switch the order.


shape keys


I doubt you use shape keys because you cannot use mirror modifiers with shape keys. But just in case, check this out. Briefly explained, you can use "mirror shapekeys" at the black triangle below the add-shapekey-button.


setting keyframes


I believe you have nothing set as a mirror object (of mirror modifier).
Add an empty and set this as the mirror object. 

As for the last solution, I made an example .blend file. 


hope it helps